Non-convex functionals penalizing simultaneous oscillations along two independent directions: structure of the defect measure

We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in {\mathbb{R}}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure ${\mathrm{\nabla }}_1{\mathrm{\nabla }}_{2}u$ of functions with finite energy. The energies depend on a parameter $\theta \in (0,1]$ and the set of functions with finite energy grows with $\theta$. For $\theta <1$ we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable in ${\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2$. We first get the result for $n_1=n_2=1$ and deduce the general case through slicing using White’s rectifiability criterion. When $\theta =1$ the situation is less clear as measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. We show however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation.